I have hole punches. I'm not sure how big my biggest round hole punch is, though. Maybe 1/2"? This leather is on the thick side, for washers, but you could do it. Getting the small hole right in the center of a disc is surprisingly difficult.
Could you drill an oversized disc with a hole saw bit for wood first to get the center hole? Then the leather punch at uniform distance from the ragged drill edge? (Maybe make a hole saw with a small center bit so that hole can be punched larger ans clean?